We need the following auxiliary statement proved by Böröczky in [1]} (see also [2]}). Let \(r:\mathbb {R}^d\setminus \lbrace {\bf 0}\rbrace \rightarrow S^{d-1}\) , \(r({\bf x})={\bf x}/\left|{\bf x}\right|\) , denote the radial projection onto \(S^{d-1}\) . We will call a \((d-1)\) -simplex a simplex in \(\mathbb {R}^d\)  with \(d\)  vertices having dimension \((d-1)\) .
